David A. Herron (University of Cincinnati)
Quasiconformal deformations and volume growth
Abstract.
Bonk-Heinonen-Koskela have produced a uniformization theory which, roughly speaking, provides a one-to-one correspondence between (certain equivalence classes of) uniform spaces and proper geodesic Gromov hyperbolic metric spaces. However, their uniformization produces an associated measure which, in general, has exponential volume growth. We determine when a metric measure space admits a uniformizing conformal density which has Ahlfors regular volume growth. Under certain mild hypotheses, this occurs precisely when the conformal Assouad dimension of the Gromov boundary is small enough.

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